to come with a legal visa and first
work at a University before they branch out. They are some good opportunities, but it
takes time to find them.

Yes. It is important to note that Universities still turn a blind eye to extra-curricula teaching by their FTs.
Be sure to follow the Chinese "elephant in the room" communication rule: Do Not ask the FAO department for permission to teach elsewhere. If you ask they must say no. They already know that all their teachers are free-lancing, but they cannot give you direct permission.